Math 3000 Lectures 18 & 19


A binary operation on a set S is a function from S x S S.

A field is a set F together with two binary operations, called addition and multiplication, which satisfy the following axioms:

1. F under addition is an abelian group.
This means that
2. F - {0} under multiplication is an abelian group.
This means that
3. a, b, cF we have a(b + c) = ab + ac (The distributive law of multiplication over addition).
4. 0a = a0 = 0 aF.
Examples of fields are given by the reals, the rationals, the complex numbers and the integers modulo p where p is a prime.

Ordered Fields

A field F is ordered if there is a relation < on F such that for all x, y, z F, Examples of ordered fields are given by the reals and the the rationals. The complex numbers and the finite fields can not be ordered.


Let A be a subset of an ordered field F. We say that uF is an upper bound for A iff a u for all aA. If A has an upper bound, A is bounded from above. Likewise, lF is a lower bound for A iff la for all aA. A is bounded from below if any lower bound for A exists. The set A is bounded iff A is both bounded from above and bounded from below.

Let A be a subset of an ordered field F. We say that sF is a least upper bound or supremum of A in F iff

Likewise, iF is called a greatest lower bound or infimum of A in F iff In some fields bounded sets may not have sup's or inf's. For example in, the set {x | x2 < 2} is bounded but does not have a least upper bound nor a greatest lower bound in.

An ordered field F is complete iff every nonempty subset of F that has an upper bound in F has a supremum that is in F.

The rationals are not complete, but the reals form a complete ordered field. In fact, it can be shown that any complete ordered field is just a copy (with the elements renamed) of the reals, so in this sense there is only one complete ordered field.

We will now give the definitions that permit us to state three important theorems about the reals.

For any real number a, if is a positive real number, the -neighborhood of a is the set N(a, ) = {x : | x - a | < } = (a -, a +).

For a set A, a point x is an interior point of A iff there exists a > 0 such that N(a, )A. The set A is open iff every point of A is an interior point of A. The set A is closed iff its complement is open.

Let B be a set. A collection of setsis a cover of B iff B is contained in the union of all the sets of. A subcover offor B is a subcollection of the sets ofwhich also cover B.

A set A is compact iff every cover of A by open sets has a finite subcover.

An element x is an accumulation point of the set A iff for all > 0, N(x,) contains a point of A distinct from x.

The set of accumulation points of a set A is called the derived set of A and denoted A'.

A sequence is a function whose domain is. We shall be talking only about real sequences, which are sequences whose codomain is.

A sequence is bounded iff its range is a bounded subset.

A sequence {xn} has a limit L, or {xn} converges to L, iff for every real> 0, there exists a natural number N such that if n > N, then | xn - L | <. If no such L exists, the sequence diverges.

A sequence {xn} is increasing iff for all n, m, xnxm whenever n < m. A decreasing sequence requires xn xm whenever n < m. A monotone sequence is one that is either increasing or decreasing.

Now we are ready to state the three theorems. One can use the completeness of to prove the famous:

Heine-Borel Theorem: A subset A ofis compact iff A is closed and bounded.

This theorem can then be used to prove the:

Bolzano-Weierstrass Theorem: Every bounded infinite subset ofhas an accumulation point in.

In turn this theorem is used to prove the:

Bounded Monotone Sequence Theorem: Every bounded monotone sequence has a limit L in.

Finally, it can be shown that if an ordered field satisfies the Bounded Monotone Sequence Theorem then the field is complete.

The circular series of proofs of these theorems shows that all of them are equivalent to the completeness of.